Planning Neural Dynamics with Lie Group Embedding through Supervised Projective Manifold Learning

Researchers propose Lie Group Embedded Dynamical Neural Networks (LieEDNN), a novel neural architecture that leverages Lie group mathematics to model continuous symmetries in dynamic systems. The approach enables stable, learnable dynamics on smooth manifolds for applications in robotics, graphics, and control systems, with experimental validation on SE(3) group structures for telescopic manipulator control.

This research addresses a fundamental challenge in neural network design: incorporating geometric constraints and continuous symmetries into dynamical systems. Traditional neural ODEs operate in Euclidean space, limiting their ability to naturally represent constrained dynamics found in physical systems. LieEDNN bridges this gap by embedding Lie groups—mathematical structures that elegantly describe continuous symmetries—directly into neural network architectures.

The theoretical contribution centers on solving two critical incompatibilities. Lie groups lack additive closure properties required for standard neural operations, and their nonlinear algebra spaces violate conventional neural ODE assumptions. The authors address these through adjoint Lie group actions that induce linear mappings, translated into block-wise weight matrix constraints compatible with perceptron-based networks. This mathematical elegance enables provably stable temporal dynamics while preserving the representational power of groups like SO(3) and SE(3).

The practical implications extend to robotics and control engineering, where systems inherently operate on curved manifolds rather than flat Euclidean spaces. Telescopic manipulators, for instance, exhibit constrained motion patterns naturally described by SE(3) transformations. By encoding these geometric constraints directly into network architecture, LieEDNN promises improved sample efficiency, stability, and interpretability compared to unconstrained approaches.

Future development hinges on scaling these techniques beyond SE(3) to more complex Lie groups and demonstrating computational advantages in real-world systems. The work establishes a foundation for geometry-aware neural networks that could reshape how AI systems model physical dynamics, particularly in applications demanding both accuracy and safety guarantees.

• →LieEDNN embeds Lie group mathematics into neural networks to naturally represent continuous symmetries and constrained dynamics.
• →Novel adjoint action mechanisms enable compatible addition operations on Lie algebras through block-wise weight matrix structures.
• →Architecture provides theoretical stability guarantees for temporal neural network dynamics on curved manifolds.
• →Applications demonstrate particular value for robotics and control systems requiring SE(3) transformations and constrained motion.
• →Approach addresses fundamental incompatibility between Lie group geometry and traditional neural ODE paradigms.